grandes-ecoles 2021 Q6

grandes-ecoles · France · centrale-maths2__official Proof Direct Proof of a Stated Identity or Equality

Let $n$ be a non-zero natural number. Let $A \in \mathbb{C}_{2n}[X]$, split with simple roots, and $(\alpha_1, \ldots, \alpha_{2n})$ its roots. Show that $$\forall B \in \mathbb{C}_{2n-1}[X], \quad B(X) = \sum_{k=1}^{2n} B(\alpha_k) \frac{A(X)}{(X - \alpha_k) A'(\alpha_k)}$$

Let $n$ be a non-zero natural number. Let $A \in \mathbb{C}_{2n}[X]$, split with simple roots, and $(\alpha_1, \ldots, \alpha_{2n})$ its roots. Show that
$$\forall B \in \mathbb{C}_{2n-1}[X], \quad B(X) = \sum_{k=1}^{2n} B(\alpha_k) \frac{A(X)}{(X - \alpha_k) A'(\alpha_k)}$$

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